I am not familiar with Humulin.
However, all medication comes with a sheet of paper describing possible side effects. Check it out.
I must admit whenever I experience some illness, pain, twinge, sniffs, coughs, ... my natural instinct is to blame diabetes.
Then I remember, diabetes does not protect us from all the usual illness, pains, twinges, sniffs, coughs, ... that everyone gets.
It is possible your cough has nothing to do with the insulin or the diabetes.